Mathematical Statistics—Things to Know from Probability

Expected Value of a Random Variable and of a Function of a Random Variable

Discrete case Continuous Case

Distribution Function Technique

Example (Also shows an example derivation of the sampling distribution of a statistic)

Suppose are independent exponential () random variables. Find the distribution of .

Recall the p.d.f. and c.d.f. for an exponential random variable:

Then, the c.d.f. of is

and the p.d.f. of is

Hence, has an exponential distribution with parameter .

Variance and Covariance

Properties of Means and Variances

Note that if the Xs are independent.

Also, if the Xs are independent, .

Moment Generating Functions

The moment generating function, M(t), of a random variable X is defined for all real values of t by . The joint m.g.f. of n random variables is .

The moment-generating-function technique is another way to determine the distribution of a function of multiple random variables. (Use the definition to write out the moment generating function of the function of random variables. Then use properties of expectation to simplify the m.g.f. Finally, if you can recognize the m.g.f. as that from a common distribution (e.g., normal, binomial), then you’ve found the distribution (the m.g.f. uniquely determines the distribution).